3.1276 \(\int \frac{x^{19}}{\left (a+b x^5\right )^2} \, dx\)

Optimal. Leaf size=59 \[ \frac{a^3}{5 b^4 \left (a+b x^5\right )}+\frac{3 a^2 \log \left (a+b x^5\right )}{5 b^4}-\frac{2 a x^5}{5 b^3}+\frac{x^{10}}{10 b^2} \]

[Out]

(-2*a*x^5)/(5*b^3) + x^10/(10*b^2) + a^3/(5*b^4*(a + b*x^5)) + (3*a^2*Log[a + b*
x^5])/(5*b^4)

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Rubi [A]  time = 0.103146, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{a^3}{5 b^4 \left (a+b x^5\right )}+\frac{3 a^2 \log \left (a+b x^5\right )}{5 b^4}-\frac{2 a x^5}{5 b^3}+\frac{x^{10}}{10 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^19/(a + b*x^5)^2,x]

[Out]

(-2*a*x^5)/(5*b^3) + x^10/(10*b^2) + a^3/(5*b^4*(a + b*x^5)) + (3*a^2*Log[a + b*
x^5])/(5*b^4)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{3}}{5 b^{4} \left (a + b x^{5}\right )} + \frac{3 a^{2} \log{\left (a + b x^{5} \right )}}{5 b^{4}} - \frac{2 a x^{5}}{5 b^{3}} + \frac{\int ^{x^{5}} x\, dx}{5 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**19/(b*x**5+a)**2,x)

[Out]

a**3/(5*b**4*(a + b*x**5)) + 3*a**2*log(a + b*x**5)/(5*b**4) - 2*a*x**5/(5*b**3)
+ Integral(x, (x, x**5))/(5*b**2)

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Mathematica [A]  time = 0.0283473, size = 49, normalized size = 0.83 \[ \frac{\frac{2 a^3}{a+b x^5}+6 a^2 \log \left (a+b x^5\right )-4 a b x^5+b^2 x^{10}}{10 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^19/(a + b*x^5)^2,x]

[Out]

(-4*a*b*x^5 + b^2*x^10 + (2*a^3)/(a + b*x^5) + 6*a^2*Log[a + b*x^5])/(10*b^4)

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Maple [A]  time = 0.008, size = 52, normalized size = 0.9 \[ -{\frac{2\,a{x}^{5}}{5\,{b}^{3}}}+{\frac{{x}^{10}}{10\,{b}^{2}}}+{\frac{{a}^{3}}{5\,{b}^{4} \left ( b{x}^{5}+a \right ) }}+{\frac{3\,{a}^{2}\ln \left ( b{x}^{5}+a \right ) }{5\,{b}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^19/(b*x^5+a)^2,x)

[Out]

-2/5*a*x^5/b^3+1/10*x^10/b^2+1/5*a^3/b^4/(b*x^5+a)+3/5*a^2*ln(b*x^5+a)/b^4

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Maxima [A]  time = 1.41995, size = 73, normalized size = 1.24 \[ \frac{a^{3}}{5 \,{\left (b^{5} x^{5} + a b^{4}\right )}} + \frac{3 \, a^{2} \log \left (b x^{5} + a\right )}{5 \, b^{4}} + \frac{b x^{10} - 4 \, a x^{5}}{10 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/(b*x^5 + a)^2,x, algorithm="maxima")

[Out]

1/5*a^3/(b^5*x^5 + a*b^4) + 3/5*a^2*log(b*x^5 + a)/b^4 + 1/10*(b*x^10 - 4*a*x^5)
/b^3

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Fricas [A]  time = 0.215, size = 95, normalized size = 1.61 \[ \frac{b^{3} x^{15} - 3 \, a b^{2} x^{10} - 4 \, a^{2} b x^{5} + 2 \, a^{3} + 6 \,{\left (a^{2} b x^{5} + a^{3}\right )} \log \left (b x^{5} + a\right )}{10 \,{\left (b^{5} x^{5} + a b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/(b*x^5 + a)^2,x, algorithm="fricas")

[Out]

1/10*(b^3*x^15 - 3*a*b^2*x^10 - 4*a^2*b*x^5 + 2*a^3 + 6*(a^2*b*x^5 + a^3)*log(b*
x^5 + a))/(b^5*x^5 + a*b^4)

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Sympy [A]  time = 2.89992, size = 56, normalized size = 0.95 \[ \frac{a^{3}}{5 a b^{4} + 5 b^{5} x^{5}} + \frac{3 a^{2} \log{\left (a + b x^{5} \right )}}{5 b^{4}} - \frac{2 a x^{5}}{5 b^{3}} + \frac{x^{10}}{10 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**19/(b*x**5+a)**2,x)

[Out]

a**3/(5*a*b**4 + 5*b**5*x**5) + 3*a**2*log(a + b*x**5)/(5*b**4) - 2*a*x**5/(5*b*
*3) + x**10/(10*b**2)

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GIAC/XCAS [A]  time = 0.233083, size = 90, normalized size = 1.53 \[ \frac{3 \, a^{2}{\rm ln}\left ({\left | b x^{5} + a \right |}\right )}{5 \, b^{4}} + \frac{b^{2} x^{10} - 4 \, a b x^{5}}{10 \, b^{4}} - \frac{3 \, a^{2} b x^{5} + 2 \, a^{3}}{5 \,{\left (b x^{5} + a\right )} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/(b*x^5 + a)^2,x, algorithm="giac")

[Out]

3/5*a^2*ln(abs(b*x^5 + a))/b^4 + 1/10*(b^2*x^10 - 4*a*b*x^5)/b^4 - 1/5*(3*a^2*b*
x^5 + 2*a^3)/((b*x^5 + a)*b^4)